Computing Discrete Logarithms with the General Number Field Sieve

The diiculty in solving the discrete logarithm problem is of extreme cryptographic importance since it is widely used in signature schemes, message encryption, key exchange, authentication and so on ((15], 17], 21], 29] etc.). The General Number Field Sieve (GNFS) is the asymptotically fastest known method to compute discrete logs mod p 18]. With the rst implementation of the GNFS for discrete logs by using Schirokauer's improvement 27] we were able to show its practicability 31]. In this report we write about a new record in computing discrete logarithms mod p and some experimental data collected while nishing the precomputation step for breaking K. McCurley's 129{digit challenge 10].